Selecting the Display Format

You can tell your TI-89 to display results in
rectangular or polar form by setting the mode (below). But however you
set your calculator to display results, you can always
enter expressions in
rectangular form,
polar form or a
mixture.

Rectangular Display Mode

Rectangular mode means you want answers in
a+bi form, whether you use polar or rectangular form when entering
your expressions.

Once only, you need to tell the TI-89 that you want
results in rectangular mode.

For complex numbers in rectangular form, the other mode
settings don’t much matter.

Polar Display Mode

“Polar form” means that the complex number is expressed as an
absolute value or modulus r and an angle or argument θ.
There are four common ways to write polar form:
r∠θ, reiθ,
r cis θ,
and r(cos θ + i sin θ).

Polar mode on your calculator means that
you want answers in a polar form,
even if you enter expressions in rectangular form.
Here’s how to set polar mode for display:

Rectangular Form for Input

Remember to distinguish between the negative-number key
[(-)] and the subtract key [−]. Use the
subtract key for numbers with interior minus like 7−3i
and 2i−11; use the
negative-number key for numbers with leading minus like
−2i and −7+3i.

Entering Expressions

Even though a complex number is a single number, it is written as an
addition or subtraction and therefore you need to
put parentheses around it for practically any operation.
The illustration shows correct methods for subtraction,
multiplication, division, and squaring.

(This screen shot was made on a TI-84, but
the TI-89 produces identical results.)

Try these operations without parentheses and you’ll see that you
get wrong answers.

Polar Form for Input

Here’s how to enter the number 4∠120°
or 4e120°i in your
calculator. Note that 120° = 2π/3 radians.

Overview:

(r ∠ θ) with angle in degrees or
radians depending on calculator mode; the parentheses are required.

Details:

Enter the absolute value or modulus, r.

[(] 4

Caution: Parentheses are required,
even if the complex number is not used in an expression.

Enter the separator between r and θ.

[2ndEEmakes∠]

Enter the angle or argument, θ.

If the calculator is in degree mode, enter the angle in
degrees, 120. The degree sign is optional if the
calculator is in degree mode.

If the calculator is in radian mode, either enter the
angle in radians, 2 [2nd^makesπ] [÷]
3, or enter the angle in degrees with a degree sign,
120 [2nd|makes°].

Enter the closing parenthesis.

[)] [ENTER]

Here’s what you get if you enter the same number when the
TI-89 is set for rectangular (a+bi)
display.
(The TI-89 panel at right shows
both exact and approximate answers.)

(You could also use re^(θi) for
entry in polar form,
but only if the calculator is in radian mode. Since the
(r∠θ) form can be used in either
degree or radian mode, I recommend you use it always.)

Conversions

Converting to Polar or Rectangular Form

Your TI-89 will automatically convert all answers to polar
or rectangular form, depending on how you set the
display format. But you can convert a
particular answer without changing the mode. The conversion command
(to Rect or to Polar) comes at the end of the command line, never in
the middle.

Finding the Angle

You can find just the angle (or argument) for a complex
number. The angle will be in radians or degrees, according to the
calculator mode.

Example: What’s the angle for the complex number
−16+47i? To
begin with, since the number is in quadrant 2 (negative real part,
positive imaginary part), the angle must be between 90° and
180° or between about 1.7 and 3.1 radians.

Select the angle function.

[2nd5makesMATH] [5] [4]

Enter the number.

[(-)] 16 [+] 47
[2ndCATALOGmakesi]

Enter the closing parenthesis and find the answer, about
108.8° or 1.8989 radians depending on your calculator mode.

(This screen shot was made on a TI-84, but
the TI-89 produces identical results.)

[)] [◆] [ENTER]
The illustration at right is shown in approximate mode. You could
use exact mode by omitting the [◆], but that's not
useful for most number.

Finding the Absolute Value r

Let’s find r, the absolute value or modulus, of the number
−16+47i.

Select the abs function.

[2nd5makesMATH] [1] [2]

Enter the number.

[(-)] 16 [+] 47
[2ndCATALOGmakesi]

Enter the closing parenthesis and find the answer, about
49.649.

(This screen shot was made on a TI-84, but
the TI-89 produces identical results.)